Exact Solution of Interacting Dissipative Systems via Weak Symemtries

In open quatnum systems, the tight connection between symmetries and conserved quantities no longer holds due to dissipation. Nevertheless, here we demonstrate how the presence of continuous weak symmetry can be used to analytically diagonalize the Liouvillian of a class Markovian dissipative systems with arbitrary strong interactions or nonlinearity. This enables an exact description of the full dynamics and dissipative spectrum. Our method can be viewed as implementing an exact, sector-dependent mean-field decoupling, or alternatively, as a kind of quantum-to-classical mapping. Although our method is general, we focus on two canonical examples: a nonlinear bosonic mode subject to incoherent loss and pumping, and an inhomogeneous quantum Ising model with arbitrary connectivity and local dissipation.We also discuss how our method provides a useful starting point to study systems where the weak symmetry is broken.